7. Let A = b3 12 and b, = a. Find A-1, and use it to solve the four equations Ax = b,, Ax = b2, Ax = b3, Ax b4 %3D
Percentage
A percentage is a number indicated as a fraction of 100. It is a dimensionless number often expressed using the symbol %.
Algebraic Expressions
In mathematics, an algebraic expression consists of constant(s), variable(s), and mathematical operators. It is made up of terms.
Numbers
Numbers are some measures used for counting. They can be compared one with another to know its position in the number line and determine which one is greater or lesser than the other.
Subtraction
Before we begin to understand the subtraction of algebraic expressions, we need to list out a few things that form the basis of algebra.
Addition
Before we begin to understand the addition of algebraic expressions, we need to list out a few things that form the basis of algebra.
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b.
0.
-4
a.
1 -2 -1
-1
, if it exists.
2. Find the inverse of the matrix A =
5 -4
2.2 EXERCISES
Find the inverses of the matrices in Exercises 1-4.
In Exercises 9 and 10, mark each statement Tr
each answer.
1. [ ]
2.
9. a. In order for a matrix B to be the
equations AB = I and BA = I must.
b. If A and B are n xn and invertible, t
inverse of AB.
3.
4.
and ab - cd +0,
C. If A =
5. Use the inverse found in Exercise 1 to solve the system
d. If A is an invertible n xn matrix,
Ax = b is consistent for each b in R
8x + 6x2 = 2
5x1 + 4x2 = -1
e. Each elementary matrix is invertible.
6. Use the inverse found in Exercise 3 to solve the system
10. a. If A is invertible, then elementary 1
reduce A to the identity In also reduc
7x1 + 3x2 = -9
-6x1 - 3x2 = 4
b. If A is invertible, then the inverse of
c. A product of invertible n xn matric
the inverse of the product is the prod
in the same order.
7. Let A =
b, =
b, =
ba =
d. If A is an nxn matrix and Ax =
every je {1, 2,...,n}, then A is
e1,..., en represent the columns of t
e. If A can be row reduced to the ide-
and b, =
a. Find A-1, and use it to solve the four equations
Ax = b1,
Ax = b2, Ax= b3,
Ax = b4
must be invertible.
b. The four equations in part (a) can be solved by the
same set of row operations, since the coefficient ma-
trix is the same in each case.
tions in part (a) by row reducing the augmented matrix
[A b b ba ba].
11. Let A be an invertible n xn matrix, ar
matrix. Show that the equation AX =
tion A- B.
Solve the four equa-
12. Use matrix algebra to show that if A
satisfies AD =I, then D = A.
13. Suppose AB = AC, where B and C ar
A is invertible. Show that B = C. Is
when A is not invertible?
8. Suppose P is invertible and A = PBP-. Solve for B in
terms of A.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fee22cf2f-b974-4b00-a3cf-09b388e7d65d%2Fd3798d55-545b-4be0-94d5-6c88bed95347%2Fv2hwiw4_processed.jpeg&w=3840&q=75)
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